A tabu search heuristic based on k-diamonds for the weighted feedback vertex set problem

Given an undirected and vertex weighted graph G = (V,E,w), the Weighted Feedback Vertex Problem (WFVP) consists of finding a subset F ⊆ V of vertices of minimum weight such that each cycle in G contains at least one vertex in F. The WFVP on general graphs is known to be NP-hard and to be polynomially solvable on some special classes of graphs (e.g., interval graphs, co-comparability graphs, diamond graphs). In this paper we introduce an extension of diamond graphs, namely the k-diamond graphs, and give a dynamic programming algorithm to solve WFVP in linear time on this class of graphs. Other than solving an open question, this algorithm allows an efficient exploration of a neighborhood structure that can be defined by using such a class of graphs. We used this neighborhood structure inside our Iterated Tabu Search heuristic. Our extensive experimental show the effectiveness of this heuristic in improving the solution provided by a 2-approximate algorithm for the WFVPon general graphs

Common adversaries form alliances: modelling complex networks via anti-transitivity

Anti-transitivity captures the notion that enemies of enemies are friends, and arises naturally in the study of adversaries in social networks and in the study of conflicting nation states or organizations. We present a simplified, evolutionary model for anti-transitivity influencing link formation in complex networks, and analyze the model's network dynamics. The Iterated Local Anti-Transitivity (or ILAT) model creates anti-clone nodes in each time-step, and joins anti-clones to the parent node's non-neighbor set. The graphs generated by ILAT exhibit familiar properties of complex networks such as densification, short distances (bounded by absolute constants), and bad spectral expansion. We determine the cop and domination number for graphs generated by ILAT, and finish with an analysis of their clustering coefficients. We interpret these results within the context of real-world complex networks and present open problems

On iterated torus knots and transversal knots

A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transversal knot in the standard contact structure for S^3 is defined to be transversally simple if it is characterized up to transversal isotopy by its topological knot type and its self-linking number. Theorem 2 of Birman and Wrinkle [op cit] establishes that exchange reducibility implies transversally simplicity. The main result in this note, establishes that iterated torus knots are exchange reducible. It then follows as a Corollary that iterated torus knots are transversally simple.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper21.abs.htm

Zero forcing in iterated line digraphs

Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks. In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications